They had some striking wallpaper in France 70 years ago. There was wallpaper that emulated the ornate gold-on-red arabesques of the Empire style, that pastiched stonework, and, far in advance of the date you might guess, paper in an assortment of modern abstract designs - coloured bars, teardrop-like blue petals, cubist triangles. You could buy wallpaper printed with ridged relief maps of the continents, olive-brown against grey oceans. Presumably, this didactic geographical wallpaper was intended for a child's bedroom. It took the imagination of Picasso to turn it into a dress.

In 1938, the 20th century's greatest painter made a work of art out of wallpaper. Women at Their Toilette is nearly four-and-a-half-metres wide and three metres tall. It towers over you, just as the history paintings in the Louvre - Gericault's Raft of the Medusa, or David's Intervention of the Sabine Women - do. Yet where these colossal paintings have content to match their scale, and are made in that distinguished medium, oil paint, Women at Their Toilette depicts a group of women beautifying themselves, and is a deliberately ludicrous confection of cut-up coloured papers.

I'd never seen more than this in Women at Their Toilette until the other day. But now it has been dramatically re-presented in a way that makes you notice its complexity and brilliance. This big, important and enigmatic work belongs to the French state as part of its huge collection of Picasso's Picassos (the works he kept with him all his life and never sold), received in lieu of inheritance tax after his death in 1973 and now displayed in the Musée Picasso in Paris. The painting has become one of the highlights of a rehang that amounts to a revision of how we see Picasso himself.

Look again at Women at Their Toilette. One woman holds a mirror to her friend who sits cross-legged; the sitting woman's face is pink and red, made of rough triangles, yet her reflection is blue and beautiful in its spiralling composition of graceful, blossoming, pastoral wallpaper. Another woman stands looking at flowers in a vase; it's when you look at her dress that you first start to suspect the meaning of Women at Their Toilette may be insidiously sombre. The dress is a map of the world: as the woman stands bolt upright, the fragments of a world map tumble downwards, one over the other - Africa, Europe, Asia and the Americas, falling into a jumbled void.

Then you look from this woman's disconcerting dress, with its images of a world hurtling to destruction, to the flowers on a high table to which she gives her attention. Picasso has used geometric designs that may indicate flower buds but also contain stars, one of which is yellow. The flower-stars make me see the Star of David. Picasso's outsized collage was made in 1938, the date of Germany's seizure of Austria and some of the first internationally visible scenes of Nazi anti-semitism in the streets of Vienna.

Start to see it like this, and Women at Their Toilette is suddenly not, after all, a lighter work than Guernica, which Picasso had painted the previous year. It is a continuation of Picasso's impassioned commentary on his times. Its daring is to capture the unreality of Europe on the eve of the second world war, and the folly of appeasement. Everything seems normal. Real life goes on. Picasso's homeland Spain may have been torn apart, but here in his adopted France the same stylish life persists, gradually becoming a nightmare. The woman sitting looking in the mirror has black skeletal arms. She's as deluded as a politician returning from Munich with Hitler's promise of peace.

The fashion for museums to keep neurotically changing their displays can be silly, but I suppose its justification must lie in turning up nuggets like Women at Their Toilette. Although it has always been on display at the Musée Picasso, the museum's new approach, in which the work has pride of place at the end of a long vista, exposes its significance in a way I'd never seen before. But then, this is no ordinary museum. The Musée Picasso gives everyone access to what was essentially the painter's diary of his creative life.

Few artists manage to shape the way their art is exhibited and seen after they are dead. Turner did, by leaving a huge portion of his work to the nation. The Picasso case is slightly different. Picasso said he was "the greatest collector of Picassos in the world"; no one knew how true this was until he died. His personal collection was a working archive, a diary of his creative life. He could go back to any of his different "periods" - the blue period, or his flirtation with surrealism - and analyse not just the general qualities of his style at that time but its detailed development. If he wanted to know what he was like as an artist, say, in the summer of 1927, he had only to look at the relevant paintings, sculptures, prints or drawings.

Women at Their Toilette is the climax to what might be called the Apollonian portion of Picasso's career. The museum's rehang divides Picasso's creativity into Apollonian and Dionysian modes - the rational and godlike giving of form, versus the bestial and savage assault on form. Picasso is the only universal artist produced by the modern age who can make us see the dread and cruelty of war, the pleasure and pain of desire, the physical wonder of everyday life. Matisse is great if you want beauty all the time; but, like a Persian rug, he won't give you crucifixions. Picasso will, and the Musée Picasso has his version of Grünewald's Crucifixion to prove it.

In separating out these two sides of the artist, the Musée Picasso has discovered a division - the divide in the Minotaur between man and beast, between Picasso the god and monster. Walk up the museum's spectacular stone stairs to the top floor and you'll find Picasso the god, whose cubist way of seeing takes the world apart, puts it back together, and makes us aware of how much experience we deny when we falsely imagine the world as a renaissance-invented perspective picture. Downstairs, in the stone vaults of the mansion, like Sadeian secrets, are Picasso's less wholesome, less edifying works of the 1920s and 30s, which disfigure the human form not to reveal its complexity, but to acknowledge how much Picasso's perception, especially of women, is distorted by rage and lust. Anger and eroticism are the same thing in grossly carnal images of faces pressed together in cannibalistic kisses, bathers throwing stones, bulls goring horses.

There is a price to be paid for emphasising works that today still shock. Abandoning its previous biographical display makes the museum a bit less accessible. But it's worth the loss of charm to show us that Picasso doesn't belong to the past. He is, more than 30 years after his death, contemporary. To make this clear the museum devotes more of its space than ever before to Picasso's use of found materials - his joyous exploitation of bits of the world, from string to sand, cigarette packets to wallpaper.

Picasso and his collaborator Braque were the inventors of collage, adding bits of cut-up paper to their cubist compositions from 1912. That same year, Picasso started to make three-dimensional assemblages. Yet somehow this revolutionary achievement has been eclipsed in the banalisation that passes for a popular history of modern art. Everyone now "knows" Marcel Duchamp invented something called "the readymade". This reductive account of how art exploded off the canvas into the world of things is pernicious, because it legitimates the crude contemporary understanding of conceptual art as, essentially, putting a toilet in a gallery. Picasso is a conceptual artist who thought with the body, and we urgently need to remember his alternative tradition of the readymade. Although examples of Picasso's constructions can be seen in other museums, many of his most experimental papiers collés and assemblages stayed in his own collection, and are here in the Musée Picasso. Its new hang gives these works star billing, from his 1912 Guitar Made of Cardboard and String, to his surrealist reliefs saturated in sand. You may know Picasso as a painter and sculptor. It's time to discover Picasso the genius of the found.

But at the heart of this 21st-century Picasso the colossal wallpaper collage Women at Their Toilette still looms. It is everything you might want art to be now: making use of found materials, ludicrous and self-mocking. And yet it goes beyond that. Picasso is not laughing at bad taste in wallpaper but parodying the grand style of history painting and alluding, in a great tease, to French portrayals of the feminine, from Delacroix's Women of Algiers to Seurat's Young Woman Powdering Herself. Beyond this travesty of gender as it has been constructed by French art, he is making deeper, more nightmarish allusions. The maps and stars speak of a world whose fine decor will soon prove paper thin. At first sight pleasantly contemporary, Women at Their Toilette soon gives way to something more menacing and disruptive. Even when he is mucking about with wallpaper, Picasso is profound.